10.3 Ratios, Proportions, Unit Rates & Percentage Calculations

Ratios, Proportions, Unit Rates & Percentage Calculations

Quick Answer: The WEST-B Mathematics subtest (Objective 0013) frequently tests proportional reasoning and percentage calculations in contextual real-world scenarios. A ratio compares quantities ($a:b$ or $\frac{a}{b}$), distinguishing between part-to-part (e.g., boys to girls) and part-to-whole (e.g., boys to total students). A proportion states that two ratios are equivalent ($\frac{a}{b} = \frac{c}{d}$) and is solved by cross-multiplying ($ad = bc$). Unit rates standardize comparisons to a single unit (e.g., cost per ounce). In percentage problems, use the foundational relationship $\text{Part} = \text{Percent} \times \text{Whole}$ ($A = P \times B$). Percent change is calculated strictly over the original base ($\frac{|\text{New} - \text{Old}|}{\text{Old}} \times 100\%$). Successive discounts must be applied sequentially (a $20\%$ discount followed by $10\%$ off yields a net $28\%$ discount, not $30\%$), and simple interest is governed by $I = Prt$.

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1. Ratios: Part-to-Part vs. Part-to-Whole

A ratio is a quantitative comparison of two numbers by division. Ratios can be expressed in three equivalent notations: $a \text{ to } b$, $a:b$, or $\frac{a}{b}$.

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|                             PART-TO-PART VS. PART-TO-WHOLE ARCHITECTURE                           |
|                                                                                                   |
|   Scenario: A classroom has 12 boys and 18 girls (Total students = 30).                           |
|                                                                                                   |
|   1. PART-TO-PART RATIOS:                                                                         |
|      - Boys to Girls:  12 : 18  -->  Simplify by GCF(6)  -->  2 : 3                               |
|      - Girls to Boys:  18 : 12  -->  Simplify by GCF(6)  -->  3 : 2                               |
|                                                                                                   |
|   2. PART-TO-WHOLE RATIOS:                                                                        |
|      - Total Parts = 2 + 3 = 5 equal parts.                                                       |
|      - Boys to Total:   12 / 30 = 2 / 5 (40% of class)                                            |
|      - Girls to Total:  18 / 30 = 3 / 5 (60% of class)                                            |
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Scaling Ratios to Find Unknown Totals

When given a ratio and a total quantity:

1. Sum the parts of the ratio to find the total number of ratio parts.
2. Divide the total quantity by the sum of parts to find the value of one part.
3. Multiply the value of one part by each individual ratio component.

Worked Example: Three-Part Ratio Problem

In a school music program, the ratio of students playing strings to brass to woodwinds is $5 : 3 : 2$. If there are 160 total students in the program, how many more students play strings than woodwinds?

1. Sum parts: $5 + 3 + 2 = 10$ parts.
2. Value of 1 part: $\frac{160}{10} = 16$ students.
3. String players: $5 \times 16 = 80$ students.
4. Woodwind players: $2 \times 16 = 32$ students.
5. Difference: $80 - 32 = \mathbf{48}$ students (or $(5 - 2) \times 16 = 3 \times 16 = 48$).

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2. Rates, Unit Rates & Comparative Pricing

A rate is a specialized ratio comparing two quantities with different units of measure (e.g., miles per hour, dollars per pound, words per minute).

The Unit Rate

A unit rate is a rate in which the denominator is simplified to exactly 1 unit of measurement:

$$\text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Total Units}}$$

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|                             COMPARATIVE UNIT PRICE (BEST BUY) ANALYSIS                            |
|                                                                                                   |
|   Option A: 24 oz jar of peanut butter for $4.56                                                  |
|   Unit Price A = $4.56 / 24 oz = $0.190 per oz                                                    |
|                                                                                                   |
|   Option B: 32 oz jar of peanut butter for $5.76                                                  |
|   Unit Price B = $5.76 / 32 oz = $0.180 per oz                                                    |
|                                                                                                   |
|   Conclusion: Option B is the 'Best Buy' (cheaper by $0.010 per ounce).                           |
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Rate of Motion (Speed, Distance, and Time)

$$\text{Distance} = \text{Rate} \times \text{Time} \quad (d = rt) \implies r = \frac{d}{t}, \quad t = \frac{d}{r}$$

• Average Speed Trap: If a car travels 60 miles at 30 mph and returns 60 miles at 60 mph, the average speed is not $\frac{30+60}{2} = 45\text{ mph}$.
  • Total distance = $60 + 60 = 120\text{ miles}$.
  • Time out = $\frac{60}{30} = 2\text{ hours}$; Time back = $\frac{60}{60} = 1\text{ hour}$. Total time = $3\text{ hours}$.
  • $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{120\text{ miles}}{3\text{ hours}} = \mathbf{40\text{ mph}}$.

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3. Proportions & Proportional Reasoning

A proportion is an equation stating that two rational expressions or ratios are equal:

$$\frac{a}{b} = \frac{c}{d} \quad (b \neq 0, d \neq 0)$$

The Cross-Product Property

In any true proportion, the product of the extremes equals the product of the means:

$$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$$

Solving Proportions:

$$x = \frac{b \cdot c}{a}$$

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|                             PROPORTION SETUP PROTOCOL FOR WORD PROBLEMS                           |
|                                                                                                   |
|   Rule: Maintain strict unit consistency across both sides of the equation!                       |
|                                                                                                   |
|          [Unit A in Numerator]      [Unit A in Numerator]                                         |
|          ---------------------  =  ---------------------                                          |
|         [Unit B in Denominator]    [Unit B in Denominator]                                        |
|                                                                                                   |
|   Example: If 3 gallons of paint cover 1,050 sq ft, how many gallons cover 2,450 sq ft?          |
|                                                                                                   |
|          3 gallons       x gallons                                                                |
|        ------------- = -------------  ==>  1,050x = 3 × 2,450 = 7,350                             |
|        1,050 sq ft     2,450 sq ft    ==>  x = 7,350 / 1,050 = 7 gallons                          |
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4. Percentage Fundamentals & Core Equations

Percent literally means "per one hundred" ($P\% = \frac{P}{100}$). All percent calculations stem from the fundamental relationship:

$$\text{Amount (Part)} = \text{Percent (in decimal form)} \times \text{Base (Whole)} \quad \implies \quad A = P \times B$$

Alternatively, using the proportional model:

$$\frac{\text{Part (is)}}{\text{Whole (of)}} = \frac{\text{Percent}}{100}$$

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|                                 THE THREE TYPES OF PERCENT PROBLEMS                               |
|                                                                                                   |
|   1. FINDING THE PART (A):                                                                        |
|      "What is 35% of 240?"                                                                        |
|      A = 0.35 × 240 = 84                                                                          |
|                                                                                                   |
|   2. FINDING THE PERCENT (P):                                                                     |
|      "45 is what percent of 180?"                                                                |
|      P = A / B = 45 / 180 = 0.25 = 25%                                                            |
|                                                                                                   |
|   3. FINDING THE WHOLE / BASE (B):                                                                |
|      "72 is 60% of what number?"                                                                 |
|      B = A / P = 72 / 0.60 = 120                                                                  |
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5. Real-World Percentage Applications

A. Percent Increase and Percent Decrease

Percent change measures the relative variation compared to the original (starting) value:

$$\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$$

WEST-B Gold Rule: The denominator of the percent change formula is ALWAYS the original starting value, never the new or final value!

• Example (Percent Increase): Enrollment grows from $4,500$ to $5,220$. $$\text{Percent Increase} = \frac{5,220 - 4,500}{4,500} \times 100\% = \frac{720}{4,500} \times 100\% = 0.16 \times 100\% = \mathbf{16\%}$$
• Example (Percent Decrease): A laptop price drops from $\text{\textdollar}800$ to $\text{\textdollar}620$. $$\text{Percent Decrease} = \frac{800 - 620}{800} \times 100\% = \frac{180}{800} \times 100\% = 0.225 \times 100\% = \mathbf{22.5\%}$$

B. Markups, Sales Tax, and Single Discounts

• Discount (Sale Price): $\text{Sale Price} = \text{Original Price} \times (1 - d)$.
  • An item listed at $\text{\textdollar}90$ on sale for $20\%$ off costs $\text{\textdollar}90 \times (1 - 0.20) = \text{\textdollar}90 \times 0.80 = \text{\textdollar}72$.
• Sales Tax (Total Cost): $\text{Total Cost} = \text{Price} \times (1 + t)$.
  • An item costing $\text{\textdollar}72$ with $8\%$ sales tax costs $\text{\textdollar}72 \times (1 + 0.08) = \text{\textdollar}72 \times 1.08 = \text{\textdollar}77.76$.

C. Successive (Compound) Discounts Trap

When multiple discounts are applied consecutively, never add the percentages together! Each successive discount applies to the newly reduced price.

Worked Example: Successive Discounts & Tax

A graphing calculator lists for $\text{\textdollar}120$. It is on sale for $25\%$ off. A teacher uses a coupon for an additional $10\%$ off the sale price. If the sales tax is $8\%$, what is the final cost?

1. First discount ($25\%$ off): $$\text{Price}_1 = \$120 \times (1 - 0.25) = \$120 \times 0.75 = \$90.00$$
2. Second discount ($10\%$ off $\text{\textdollar}90$): $$\text{Price}_2 = \$90 \times (1 - 0.10) = \$90 \times 0.90 = \$81.00$$ (Note: Adding $25\% + 10\% = 35\%$ would erroneously yield $\text{\textdollar}120 \times 0.65 = \text{\textdollar}78.00$).
3. Apply Sales Tax ($8\%$ on $\text{\textdollar}81$): $$\text{Total Cost} = \$81.00 \times (1 + 0.08) = \$81.00 \times 1.08 = \mathbf{\$87.48}$$

D. Simple Interest Formula

$$I = Prt$$

• $I = \text{Interest earned or paid } (\text{\textdollar})$
• $P = \text{Principal (initial amount invested or borrowed } \text{\textdollar})$
• $r = \text{Annual interest rate (expressed as a decimal, e.g., } 4.8\% = 0.048)$
• $t = \text{Time in YEARS}$ (If time is given in months, $t = \frac{\text{months}}{12}$)
• Total Future Balance ($A$): $A = P + I = P(1 + rt)$.

Worked Example: Interest for Fractional Year

A district deposits a $\text{\textdollar}25,000$ grant at $4.8\%$ annual simple interest for 9 months. Calculate interest earned.

1. Identify variables: $P = 25,000$, $r = 0.048$, $t = \frac{9}{12} = 0.75$ years.
2. Compute: $$I = 25,000 \times 0.048 \times 0.75 = 1,200 \times 0.75 = \mathbf{\$900}$$

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6. Diagnostic Error Analysis & Educator Strategies